NBER WORKING PAPER SERIES

THE CLOSING OF THE GENDER GAP AS A ROY MODEL ILLUSION

Casey B. MulliganYona Rubinstein

Working Paper 10892http://www.nber.org/papers/w1089

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138November 2004

We appreciate the comments of Gary Becker, Aitor Lacuesta, Amalia Miller, Kevin M. Murphy, JohnPepper, Ed Vytlacil, the research assistance of Ellerie Weber, and the financial support of the NationalScience Foundation (grant #0241148). The views expressed herein are those of the author(s) and notnecessarily those of the National Bureau of Economic Research.

© 2004 by Casey B. Mulligan and Yona Rubinstein. All rights reserved. Short sections of text, not to exceedtwo paragraphs, may be quoted without explicit permission provided that full credit, including © notice, isgiven to the source.

The Closing Gender Gap as a Roy Model IllusionCasey B. Mulligan and Yona RubinsteinNBER Working Paper No. 10892November 2004JEL No. J21, J31, J16, C31

ABSTRACT

Rising wage inequality within-gender since 1975 has created the illusion of rising wage equality

between genders. In the 1970's, women were relatively equal (to each other) in terms of their

earnings potential, so that nonwage factors may have dominated female labor supply decisions and

nonworking women actually had more earnings potential than working women. By 1990, wages had

become unequal enough that they dominated nonwage factors, so that nonworking women tended

to be the ones with less earnings potential, and the wage gap between workers and nonworkers was

large. Accounting for the growing selection bias using both parametric and semi-parametric versions

of the Roy model, we show how the earning power of the median woman has not caught up to the

earning power of a median man, even while the earning power of the median working woman has.

As an illustration, we give some attention to wives with advanced degrees � they have high and

stable labor force participation rates � and show how their measured wages have grown at about

the same rate as those of men with advanced degrees.

Casey B. MulliganUniversity of ChicagoDepartment of Economics1126 East 59th Street, #506Chicago, IL 60637and NBERc-mulligan@uchicago.edu

Yona Rubinsteinyonar@post.tau.ac.il

Table of Contents

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II. Selection and Composition Biases in the Calculation of Relative Wages . . . . . . . . . . . . . . . . . . . 5Labor Supply Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Labor Supply Varies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

III. Recalculating Gender Gap Closure with Heckman and Related Selection Models . . . . . . . . . . 11

IV. Further Indicators of the Composition of Working Wives, Related to Identification at Infinity 17The Composition of Advanced Degree Working Wives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Determinants of Marriage Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

VI. Appendix: The Occupational Composition of Wives with Advanced Degrees . . . . . . . . . . . . . 21

VII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1Perhaps also in small part an negative effect of womens labor market entry on unskilledmale wages(eg., Topel, 1994; Juhn and Kim, 1999).

I. Introduction

Perhaps one of the most dramatic changes in U.S. labor market outcomes over the past thirty

years is the persistent growth, within-gender, in overall earnings inequality (see Levy and Murnane,

1992, and Katz and Autor, 1999, for comprehensive surveys). Inequality in earnings grew over this

period not only from an increase in the Mincerian returns to education but also due to an increase in

inequality within groups of workers of similar age and education (Katz and Murphy, 1992). As first

pointed out by Juhn, Murphy, and Pierce (1993) the inequality growth during the 1970s, the 1980s, and

the 1990s, appears to have occurred throughout the earnings distribution as well as over people's life

cycle (Gottschalk and Moffitt, 1995). By the end of the second Millennium, US wage inequality was

higher than it has ever been since WWII.

The measured earnings of women have substantially, although not fully, caught up with the

earnings of men. This closing gender wage gap is often said to indicate the importance of wages for

bringing women into the labor force (Mincer, 1962; Goldin, 1990), and the status of discrimination and

other nonwage factors in the operation of the labor market (Becker, 1985; Katz and Murphy, 1992). At

the same time, it has been observed that this increased gender equality has been coincident with growing

earnings inequality within genders, as shown in Figure 1. The solid line is a familiar measure of gender

equality (e.g., Murphy and Welch, 1999), namely, the median earnings of women working full-time full-

year as a ratio of the median earnings of men working full-time full-year (hereafter ftfy). The dashed

line is the ratio of the 90th percentile to the 10th percentile in the cross-section distribution of men working

full-time full-year. We see that both were flat until 1977 or so (see also ONeill, 1985, on the apparent

constancy of the gender wage gap). Both rose most rapidly at first from the late 1970's until about

2000. This has been regarded in the literature as a curious coincidence (Card and DiNardo, 2002, p. 742;

Blau and Kahn, 1997, p. 2),1 and perhaps indicative of earnings having multiple and largely independent

determinants, but can we still conclude that wages help pull women into the labor force, and that the

Gender Equality 2

2Figure 2's calculations of the advanced degree gender gap are based on 400+ annual CPSobservations of married women working full-time, full-year.

labor market has mitigated gender discrimination over time? The purpose of our paper is to suggest that

(a) apparent gender equality is a direct consequence of inequality within gender, and (b) the apparent

gender equality is not real in the sense that the average womans earnings potential has not caught up

with that of the average man.

Although more work on this topic is needed, our two results may suggest a third, namely that

much of earnings inequality growth can be understood with a single attribute model. In such a model,

women would have less earnings potential than men for the same reason that some men have less

earnings potential than others: differences in the one attribute (call it skill). According to our estimates

below, the average married woman in 1975 had 36% less earnings potential than the average married

man, and would fit in the 17th percentile of the 1975 male wage distribution. That percentile lost 18%

relative to the male average over the period 1975-95, so in the single attribute model women should have

lost the same percentage. Some of our estimates do suggest that the gender wage gap widened 1975-95,

although it is unclear whether it widened that much. Nevertheless, the single attribute model performs

much less badly than it appeared when it was thought that inequality moved in opposite directions within

and across genders.

We argue that, as suggested by the Roy (1951) model as applied to the choice between market

and nonmarket work, working women are selected based on their wages, and the gap between the

earnings potential of working and non-working women grows with inequality among women. As a

consequence, the wage distribution of working women is different than the wage distribution of women,

to a degree which declines with the fraction of women who work. Hence, a less biased estimate of the

gender gap may be most easily calculated, and shown to close more slowly if at all, using samples of

women for which the propensity to work is high, and stable over time. Figure 2 displays such an estimate

as the solid line, namely the gender wage gap calculated as the average log wage for married women with

advanced degrees working ftfy, net of the average log wage of ftfy men with advanced degrees. Work

is very common for this group of women about 80% of them have some earnings during the year and

about half work ftfy so we expect earnings inequality to affect less significantly the trend in this

groups relative wages. The solid line shows little growth in the relative earnings of women. In contrast,

other schooling groups have fewer women working (the 1970 percentage of women working ftfy is

shown in parentheses in the legend), and thereby a greater potential for selection and composition biases,

and show growth in the relative earnings of women much like that shown in Figure 1.2 Indeed, while

Gender Equality 3

Figure 1 Wage Inequality between and within Genders

selection bias growth may be less important for the advanced degree women, it still may have been

positive; the solid line actually suggests that the earnings potential of women may have fallen over time

relative to mens.

Gender Equality 4

Figure 2 The Gender Gap by Schooling

Section II uses the Roy model to show how growing inequality within gender can give the

illusion of a closing gender gap, due to the selection and composition biases involved with measuring

the earnings potential of women. The Roy model predicts that a groups measured wage growth is an

upward biased measure of its earnings potential growth, with the magnitude of the bias declining with

the fraction of group members actually working. Section II also shows how a number of demographic

groups, including married women, fit this pattern. Section III uses structural selection models to obtain

numerical estimates of the amount of selection bias growth, and thereby the relative growth of male and

female potential earnings. Section IV explores alternative interpretations of our Figure 2. Section V

concludes.

Gender Equality 5

Our data come from a series of 39 consecutive March Current Population Surveys and their

Demographic Supplements (hereafter: March CPS) for survey years 1964 to 2002. The population

sample (universe) consists of civilian non-institutionalized population of the US living in housing units

and members of the Armed Forces living in civilian housing units on a military base or in housing units

not on a military base. Each record contains information about an individual, the household in which the

individual resides, and the family and the spouse of the individual. In addition to the standard monthly

labor force data, these files contain supplemental data on work experience. This collection provides

information on employment and wages in the preceding calendar year while demographic data refer to

the time of the survey. Thus, the annual work experience data from the CPS demographic supplement

cover the period of 1963 to 2001. We construct two data sets. The first file includes all individuals

aged 24 to 54 (hereafter: individual file). The second file includes only husbands and wives. We restrict

the second file to include only couples in which we observe both partners (1,248,117 couples in 1964

through 2002).

CPS observations are divided by school completion into five sub-groups: (i) high school dropouts

less than twelve grades, (ii) high school graduates (including those graduated by taking the GED

exam), (iii) some college completed, (iv) college graduates with 16 (and 17) years of schooling (BA) and

(v) college graduates with advanced/professional degree (MBA, Ph.D.) or, prior to 1993, persons with

18 or more years of completed schooling. We measure wages according to total annual earnings deflated

by the US CPI, giving most of our attention to ftfy samples (namely full-time workers who report

working at least 50 weeks of the previous year).

II. Selection and Composition Biases in the Calculation of Relative Wages

Wages are often interpreted as measures of important economic concepts, such as human capital

or discrimination. However, one nuisance in the measurement process is the difficulty of measuring

wages for people who are not working. Labor economics suggests that the people who work are different

from the people who do not work, and statistical theory shows that the wage difference between those

working and those not working increases with the amount of wage inequality. The purpose of this paper

is to show that, in the context of the rising inequality 1975 until the present, this effect is important and

explains a lot of the cross-group pattern of wage gains over that period. Perhaps most important is the

real possibility that the measured wage gains of married women relative to married men are consistent

with the view that the wage distribution for wives was actually unchanged relative to the wage

Gender Equality 6

wr

~ Nwr

,σ2w ρwrσwσr

ρwrσwσr σ2r

E(w |z > 0) ' w % λ(L)ρwzσw

L ' Φ w & rσz

ρwz /σw & ρwrσr

σz, σ2z / σ

2w % σ

2r & 2ρwrσwσr , λ(L) /

n(Φ&1(L))L

(1)

distribution for husbands.

The Roy (1951) model, as applied to the choice between market and nonmarket work (see also

Gronau, 1974, and Heckman, 1974), illustrates this. Each person is described by two variables: his or

her (potential) market wage, and his or her reservation wage (a.k.a., his or her productivity in the

nonmarket sector). A person works in the marketplace if and only if the market wage exceeds the

reservation wage. Each person's market and reservation wages are drawn from a joint lognormal

distribution, whose parameters may vary over time and across groups,

where w and r denote log market and reservation wages, respectively, and hats denote medians. The

workers L are distinguished from the nonworkers by the condition z / w - r > 0, where z is the net gain

from working. Since wages are unmeasured for nonworkers, the average measured wage is E(w|z>0):

where σz is, roughly speaking, the inverse of the group labor supply elasticity. λ is the inverse Mills

ratio, and slopes down as a function of L.

ρwz is the correlation between log wages and the (log) net gain from working, which can either

be positive or negative, according to whether workers have higher or lower wages than nonworkers,

respectively. Just as important, growth in σw should increase ρwz and could even change its sign.

Remember that σw was much lower in the 1970's, at a time when ρwz was found to be negative for married

women (e.g., Heckman, 1974). ρwz < 0 is equivalent to σw < ρwrσr, which should be less likely to hold

as σw gets larger. Indeed, we find that ρwz changes sign for married women in the early 1980's.

Intuitively, nonwage factors dominated female labor supply decisions in the 1970's when σw was

relatively small. By 1990, wages had become unequal enough that they dominated nonwage factors, so

Gender Equality 7

3As shown in the second formula (1), an increased labor supply might come from highermedian market wages, lower median reservation wages, or a change in the labor supply elasticity. The labor supply elasticity is determined by the amount of inequality in the net gain z from working.

∆E(w |z > 0) ' ∆ w % λ(t)ρ(t)∆σ % λ(t&1)∆ρσ(t&1) % ∆λρ(t)σ(t&1) (2)

that nonworking women tended to be the ones with less earnings potential.

Equation (1) decomposes the average measured log wage into four components, two of which

have been emphasized in the gender wage gap literature. The first and obvious one is the median wage.

For example declining gender discrimination is sometimes said to uniformly increase the potential market

wage of all women, perhaps as modeled by shifting the median wage. Second is a form of composition

bias emphasized by ONeil (1985), Blau and Kahn (1997), and others: when ρwz > 0, labor supply shifts

move relatively low wage people into (or, if the shift is in the direction of less labor supply, out of) the

labor market.3 The magnitude of this composition bias depends on the Mills ratio, which is higher when

a smaller fraction of the group is in the labor market. Third is another form of composition bias. In

general, at least with ρwz > 0, workers are some combination of high market wage and low reservation

wage. ρwz indicates the relative importance of these. Fourth, to the extent that workers are selected on

wages, workers have higher wages. Gronau (1974, pp. 1127-8) and others recognize that the magnitude

of the selection bias decreases with the amount of labor supply L, and increases with the amount of wage

inequality σw. However, Gronaus result has been ignored when considering wage trends since 1975,

namely when σw was growing.

II.A Labor Supply Constant

Henceforth, we refer to equation (1)s bias term without subscripts namely as λρσ except

when needed for clarity. The change over time in a groups average measured log wage has four

components corresponding to the four biases mentioned above.

where t denotes time and ∆ denotes a change from time t-1 to time t. For the time period 1975-2000, ∆σ

and ∆ρ are presumably positive, since within-group wage inequality grew during this period, and ρ

increases with σ (see equation (1)). The sign of ∆λ depends on the particular group, namely whether the

group increased or decreased its labor supply. Hence we begin by considering two groups whose labor

supply was little changed during the period, namely high school educated men aged 40-49 and high

Gender Equality 8

4Over the period shown in the Figure below (1974-78 to 1994-98), the fraction of high schooleducated men aged 40-49 working ftfy fell from 0.77 to 0.73 (authors calculation from the CPS). The fraction for high school educated men aged 66-70 fell from 0.14 to 0.11.

∆E(w |z > 0) ' ∆ w % λ(t&1)[ρ(t)∆σ % ∆ρσ(t&1)] (2)N

school educated men aged 66-70.4 For them, and any other group whose labor supply constant is

constant because changes in the median reservation wage offset changes in the market wage distribution,

λ(t) = λ(t-1) and equation (2) becomes (2)N.

In the case that workers earn more than nonworkers would (ρ > 0), the square bracket term is positive,

and growing inequality causes measured average log wages to grow more than do median log wages.

Furthermore, the magnitude of the bias is proportional to λ(t-1), which declines with group labor supply

L(t-1), and should be close to zero for groups like men aged 40-49 for whom L is practically one.

We expect gender and schooling to be the more important determinants of the ∆w term, with age

less important because most work on the age structure has shown fairly little change over time in the

returns to experience. In order to bring our attention to the bias term, we compare groups of the same

gender and schooling, and for the moment treat the ∆w as a constant. In terms of the bias term, the

second term in square brackets could be as large the first. To see this, notice that inequality as measured

by σ changes by about 0.13 1975-2000, and that ∆σ is multiplied by ρ which may be a lot less than 1,

so that the product is less than 0.13. The initial level of σ is about 0.6, and is multiplied by ∆ρ, which

from the formula (1) could be as large or larger than 0.1. Hence 0.1 or 0.15 is a reasonable guess for the

square bracket term. The group-difference in inverse Mills ratios is 1.19, so we expect, to an order of

magnitude, the group difference in ∆E(w|z>0) to be roughly 0.15.

Figure 3 shows that, empirically, the group difference is 0.15. The horizontal axis shows L(t-1).

The vertical axis shows ∆E(w|z>0) for the two groups relative to ∆E(w|z>0) = -0.255 for all prime-aged

high school educated men. The L=1 intercept in Figure 3 is important because λ(L=1) = 0 and equation

(2)Ns bias term disappears. Obviously, the selection and composition biases are zero for a group with

100% labor supply. Since the L=1 intercept is about zero, it appears that most of the measured wage

gains for elderly high school educated men (relative to high school educated men overall) may just be

an illusion, in the sense that they would be observed even if w were constant. Perhaps this conclusion

is not particularly surprising or interesting, but the technique of using the L=1 intercept leads to some

novel conclusions for women.

Gender Equality 9

Figure 3 Measured Wage Growth Declines with Labor Supply (Men)

Most groups of married women significantly increased their labor supply in the 1970's, 1980's,

and 1990's, so equation (2)N does not apply. However, this is much less true for married women with

post-college education: 43% of them worked full-time full-year in the early 1970's compared with 55%

in the late 1990's (the same percentage went from 27 to 47 for college grads and from 24 to 45 for high

school grads). Where should they appear in Figure 3? ρ is probably smaller, and even negative early

in the period, for women than for men because husbands and wives sort positively, and the husbands

wage has a negative wealth effect on wifes labor supply. In the absence of gender-specific determinants

of the ∆w term, men and womens groups with constant labor supply should have the same L=1 intercept,

but the slope of the wage growth-labor supply relation be smaller in absolute value for women. In fact,

measuring wage growth relative to men with the same schooling, we find that post-college wives would

be located at (0.43,0.06) in Figure 3. As predicted by the Roy model, their (schooling adjusted) wage

growth is low like that for husbands aged 40-49.

Gender Equality 10

∆E(w |z > 0) ' ∆ w % ρ(t) [λ(t)σ(t) & λ(t&1)σ(t&1)] % λ(t&1)∆ρσ(t&1)

. ∆ w % λ(t&1)∆ρσ(t&1)(2)O

II.B Labor Supply Varies

Labor supply trends for some groups, especially married women. For the purposes of using the

formula (2), the various sources of the labor supply trend (which may include σ, see the formula (1) for

L) all matter in the same way, namely as they contribute to changes in the inverse Mills term λ. Since

married womens labor supply is increasing, their inverse Mills term λ is decreasing and equation (2)s

∆λ term has the opposite sign as its ∆σ term. Equation (2)O rewrites equation (2) by combining these

terms:

We measure σ as the standard deviation of log annual earnings among men, and λ by applying the inverse

Mills formula to observed married female labor supply. The square bracket term turns out to be -0.14,

and is multiplied by ρ(t). Since the last term is ∆ρ times 0.71, it probably dominates the middle term for

married women, which is our reason for using the approximation shown in the bottom line of equation

(2)O. In words, measured wage growth for women is biased upward for men and, like the bias for men

aged 40-49 or 66-70, the magnitude of the bias depends on the initial level of labor supply through the

inverse Mills term λ(t-1). Whether the bias is larger for male or female groups with similar λ(t-1)

depends on the relation between ∆ρ and gender. As we explain below, we expect ∆ρ to be positive and

much larger for women.

Figure 4 is a female version of Figure 3. The horizontal axis measures group labor supply 1974-

8. The vertical axis measures group wage growth relative to men with the same schooling. Again we

see that the high initial labor supply groups (single women and/or college+ women) have lower wage

growth. The relation is steeper within marital status than across marital status, which we expect if

growing inequality has a additional wealth effect on wives (through the wages of their husbands). In any

case, the high labor supply groups in Figure 4 tell us that the gender wage gap may not have closed at

all, or closed at most 0.15, which is much less closing than suggested by the dashed and dash-dot line

in Figure 2.

Gender Equality 11

Figure 4 Measured Wage Growth Declines with Labor Supply(White Women)

III. Recalculating Gender Gap Closure with Heckman and Related Selection Models

If we modify equation (1) by allowing median reservation and market wages to be log-linear

functions of demographic characteristics X, it becomes the Heckman (1979) selection model. Remember

that the Heckman selection model can be interpreted as a least squares regression of log wages on X plus

the inverse Mills ratio λ predicted for the worker based on her demographics; conversely that least

squares regressions of log wages on X suffer from the bias resulting from the omission of the inverse

Mills ratio λ. Hence, if the relation between demographics and median wages were constant (our

estimates below suggest that it is), then an increase over time in the λσρ term causes the constant term

in the Heckman selection model to increase less (or decrease) more than the constant term in the least

squares model. We explained above how the change over time in λσρ is qualitatively ambiguous because

Gender Equality 12

λ falls and σ rises, but the Heckman selection model permits numerical estimates of λσρ. We display

some estimates in Table 1. The left part of the Table uses married women from the 1970's, and the right

part uses married women from the late 1990's. On each side, a least squares and Heckman selection

estimates are shown; of course the Heckman specifications have (or can be interpreted as having see

Heckman 1979) λ as an additional regressor.

Table 1: Womens wages over time, with and without selection corrections

1975-79 1995-99 selectionbias growth

independent variable OLS Heckit OLS Heckit

(experience-15) 0.005(0.002)

0.005(0.002)

0.016(0.002)

0.018(0.002)

(experience-15)2/100 -0.011(0.010)

-0.010(0.010)

-0.065(0.009)

-0.074(0.010)

high school dropout 9.787(0.013)

9.888(0.034)

9.686(0.019)

9.460(0.033)

0.327

high school graduate 9.999(0.013)

10.085(0.027)

9.960(0.008)

9.788(0.022)

0.258

some college 10.084(0.011)

10.166(0.027)

10.150(0.008)

9.983(0.021)

0.249

college graduate 10.282(0.013)

10.353(0.025)

10.492(0.008)

10.334(0.020)

0.228

advanced degree 10.500(0.021)

10.556(0.027)

10.774(0.012)

10.637(0.020)

0.193

teacher 0.033(0.017)

0.035(0.017)

-0.233(0.013)

-0.231(0.013)

0.000

observations 20,971 20,971 28,931 28,931

σρ 0 -0.066(0.020)

0 0.183(0.022)

adj-R2 .07 .07 .18 .18

Notes: (1) dependent variable is log weekly wage. sample is wives aged 25-54 from white households(2) there is no constant term, but the schooling dummies sum to a constant(3) selection bias growth is the growth over time of the OLS minus Heckit coefficient on the schooling dummy(4) standard errors in parentheses(5) experience measured as age - years of schooling - 6(5) Heckit model estimated in two stages, with the first stage including wifes education and experience, husbandseducation and experience, and the number of children aged 0-6 in the family

Gender Equality 13

The regressions shown in the Table have no constant term per se, although the schooling

dummies sum to one. Hence the education coefficients estimate the mean (with the normal distribution,

also the median) log wage for a nonteacher with 15 years of experience (experience measured as age

minus schooling minus six). According to the least squares estimates, some college womens median

log wages increased by 0.066 log points. Since mens wages were higher and declining over this period,

this might be interpreted as a closing of the gender gap. However, the Heckman selection estimates say

that mean log wages actually fell 0.183 log points; there was little or no gender gap closure. The reason

for the different Heckman estimates is that the inverse Mills ratio coefficient was negative during the

1970's and positive during the 1990's. In words, the bias from not measuring the earning power of

nonworking women has changed over time (for some college, by 0.249 log points), in large part

because wage inequality has grown within gender.

Figure 5 displays time series for wives log wage selection bias. More specifically, Figure 5 is

a graphical version of Table 1, with nine time periods rather than two: in each time period the Heckit

constant term (for women with some college) is subtracted from the corresponding OLS constant term.

During the 1970's, the selection bias was negative (i.e., the selection correction was positive); women

out of the labor force had more earnings potential than women in the labor force. Beginning in the early

1980's, the selection bias became positive. Overall, womens wage growth is 25-30% less when

corrected for selection. Figure 5 suggests that all of the gender gap closure shown in Figure 1 is due to

changing selection bias!

Gender Equality 14

Figure 5 Wives log wage selection bias over time

Although using different methods and concerned with wage gaps by race rather than gender, Neal

(2004) has a result analogous to our Figure 5. More precisely, while we show that the selection bias for

women is greater (and of the opposite sign) in recent decades than in the 1970's, Neal shows that the

(1990) selection bias is greater (and perhaps of the opposite sign) for black women than for white

women. Neal finds a (gender-) differential selection bias of 0.1, while we find a (time-) differential

selection bias of as much as 0.3 (see Figure 5).

Although it may be surprising to see how much changing selection bias contributes to measured

gender gap closure, it is not surprising that the sign of the selection bias might have changed over time

as suggested by Figure 5. For example, married female employment rates have increased somewhat more

among the more educated (Juhn and Kim, 1999, Table 2). In the early 1970's, the average education of

Gender Equality 15

5Selection bias growth varies little across schooling groups in all of the specifications wehave tried, so henceforth we display selection bias for the some college group without reference tothose for the other groups.

6The level of the selection bias does depend on specification. For example, the selection biasis significantly more negative (less positive) for Table 2's specification (1) than specification (2).

working wives husbands was essentially the same as that for nonworking wives. By 1990, the average

education of working wives husbands exceeds that for nonworking wives by about 1 year. The more

dramatic change in the relative quality of the married female work force is shown by Juhn and Murphy

(1997) and Juhn and Kim (1999), who stratify married women by their husbands position in the married

male wage distribution. In 1970, the employment rate of married women with husbands at the bottom

of the male wage distribution was 0.44, as compared to 0.31 for married women with husbands at the top.

By 1990, the wives employment rate was essentially independent of husbands position, for example

0.60 at the bottom and 0.61 at the top. In other words, female labor force growth seems to have come

disproportionately from skilled women (see also Topel, 1994).

It is well known that the slope coefficients in womens wage and labor supply equations are

sensitive to alternative specifications (e.g., Mroz, 1987). But what about the growth over time in the

selection bias terms? First, selection bias growth is similar for various schooling groups.5 The various

entries in Table 1's last column are selection bias growth for various schooling groups. In all cases the

selection bias growth is in the range 0.20-0.32, so that all of the measured gender gap closure for these

groups appears to be selection bias growth. Second, married women selection bias growth is not

sensitive to the reassignment of variables from the regression equation to the selection equation, or vice

versa. Table 2 displays estimates of selection bias growth for Heckit specifications that differ according

to the independent variables used in the selection and/or regression equations; six of the eight estimates

are in the range 0.21-0.41, with the two extremes as 0.05 and 0.88.6 Figure 4 does suggest some

specification sensitivity when we include single women because, for example, the selection bias for

married high school graduates appears less when we compare them with single high school grads then

when we compare them with married advanced degree women. Nevertheless, Figure 4 clearly shows

that wage growth falls significantly with the labor force participation the only question raised by that

Figure is whether all, or just half, of the measured gender gap closure is selection bias.

Gender Equality 16

7Interestingly, Newey (1999) suggests that 1-L may be more robust to changes indistributional functional forms than the other specifications.

Table 2: Selection Bias Growth fromVarious Specifications of the Regression (R) and Selection (S) Equations

independent variables (1) (2) (3) (4) (5) (6) (7) (8)

number of children 0-6 S S R, S R, S R, S S S S

husbands age S R, S R, S R, S S S

husbands education S R, S R, S S R, S S

selection bias growth 0.249 0.217 0.876 0.054 0.413 0.222 0.224 0.208

Notes: (1) each column is a different specification of the Heckit model(2) wifes experience and schooling included in all regression and selection equations(3) samples of married white women from the CPS(4) selection bias growth is the growth from 1975-79 to 1995-99, for the some college schoolinggroup, and 15 years of experience

Third, we expect quantitative, but not qualitative, results to be sensitive to wage distribution

functional form. As discussed above in relation to equation (1), Figure 3, and Figure 4, the selection bias

growth is negligible for groups of women with high and stable labor supply: a qualitative result which

does not rely on the assumed lognormal distribution. Since we see log wage growth sloping down with

the level of labor supply, we expect the high and stable labor supply groups to tell us the most about

genuine gender gap closure, even if the wage distribution were not lognormal. Of course, in order to

obtain a quantitative estimate of the selection bias growth, the Heckman two-step procedure uses the

inverse Mills ratio, with derives from normal functional form. Moffitt (1999) and Newey (1999) have

suggested using, as a robustness check on the distributional functional form, monotone transformations

of the inverse Mills ratio (equivalently, monotone transformations of the predicted probability of

working). Table 3 uses different transformations predicted probability of working and different

instrumental variables. Using 1-L as the additional regressor (see the middle column), we find the

selection bias growth to be essentially the same as with the lognormal model (see the first column).

Using both the inverse Mills ratio and 1-L, we find less selection bias growth (see the last column),

although in the same direction.7

Gender Equality 17

Table 3: Selection Bias Growth fromVarious Specifications of Distributional Functional Forms

instrumental variables λ(L) 1-L λ(L), 1-L

number of children 0-6 0.265 0.350 0.145

number of children 0-6, husbands age,and husbands education

0.249 0.267 0.090

Notes: (1) selection bias growth from 1975-9 to 1995-9, for the some college schoolinggroup. The 0.249 estimate is the same estimate as shown in the last column of Table 1(2) L denotes the predict probability of working ftfy, and λ denotes the inverse Mills ratio

IV. Further Indicators of the Composition of Working Wives, Related to Identification at Infinity

Growing inequality within gender likely contributes to the closing of the measured gender gap

because of its effect on the nature of the selection of women into the labor force. As we point out above,

the direction of the bias seems clear, as on various observable characteristics the female workforce has

improved its composition. Structural selection models are one way of calculating a numerical gender

gap that is free from selection bias. In the spirit of Chamberlins (1986) and Heckmans (1990)

identification at infinity argument, we have also proposed to focus on wage growth for groups of

women with high and stable labor supply, namely the single women and advanced degree wives featured

in our Figures 2 and 4. However, these estimates may still be biased to the extent that the composition

of groups of women with high and stable labor supply have changed over time relative to their male

comparison group. Subsection A makes two comparisons for advanced degree wives: working wives

as compared with the general population of advanced degree wives (ie., regardless of labor force status),

and advanced degree wives as compared to the general population of wives with a college diploma or

higher. Subsection B displays comparisons of married and single women.

IV.A The Composition of Advanced Degree Working Wives

It is well known that husbands and wives sort positively on many characteristics: height, race,

schooling, and (among dual-earner couples) even earnings. Our strategy here and in subsection B is to

use the earnings of a womans husband as a proxy for her own earning ability. Figure 6s solid line is

the difference between the average log weekly wages of two groups of husbands: husbands of women

who have advanced degrees and the husbands of women who have at least graduated college. The series

has trends up since 197, which suggests that the group of advanced degree women has grown via the

Gender Equality 18

Figure 6 Composition of Advanced Degree Working Wives,using Husbands Earnings as a Proxy

addition of women with relatively high earnings potential.

IV.B Determinants of Marriage Rates

Figure 4 also shows that prime-aged single women as a group: (a) supply more labor, and (b)

have enjoyed less wage growth than married women. Has single female wage growth been less because

their human capital (as a group) has declined relative to that of married women? It is well known that

prime-aged single women have become increasingly black, but that is not the explanation of Figure 4

because it uses only samples of white women. Figure 7 displays the fraction of white women, stratified

by marital status, with a college or advanced degree. Here we see that human capital has grown for both

groups in the same amount until the early 1990's. Since then, the fraction with degrees has grown only

for married women, by about eight percentage points.

Gender Equality 19

Figure 7 Fraction of White Women aged 25-54with College Degree or Higher

If declining relative human capital of single were to explain the kinds of wage growth shown in

Figure 4, then Figure 7 suggests that it would have occurred since 1990. More work comparing married

and single women is needed, but for our calculation it appears that much of the relative decline in wages

of single women occurred prior to 1990.

V. Conclusions

Growing earnings inequality has been associated with the loss in earnings potential for some

groups of people, and the growth in earnings potential of others. Wives not working in the marketplace

include a disproportionate share of people with lost earnings potential, but are not included in

calculations of female earnings. Hence, growing earnings inequality within gender has created the

illusion of a gender wage gap that closes more rapidly than the average woman has gained relative to the

Gender Equality 20

average man, if she has gained at all. How large is this bias? We offer a parametric estimate and two

simple nonparametric estimates, based on the principle that the effects of labor force selection are

minimal among groups with high and stable labor supply. The first nonparametric estimate is the gender

gap among married people with advanced degrees, which is about 0.25 or 0.30 log points throughout the

years 1963-2000. This estimate, which implies that the average married women has gained only 0 or

0.05 log points in earnings relative to the average married man, may hide further selection bias growth

and thereby hide an actual opening of the gender gap, because the employment rate of advanced degree

women is still less than 100%. Our second nonparametric estimate is the gap between the earnings of

single women and the earnings of men with similar education. The second gap closes, but only about

half as much as the raw gender gap featured in the literature.

Finally, parametric estimates of the Heckit model (assuming lognormal distributions) suggest that

the average married women did not gain, and may have even lost, relative to the average married man.

All together, it appears that the relative earnings progress of women since 1963, if any, has been limited.

Furthermore, we will see standard measures of the gender gap widen in the future, if and when earnings

inequality within genders returns to 1970's levels.

We use the Roy model to decompose the selection bias into components related to changing

labor supply, growing inequality within gender, and a changing cross-sectional correlation between

wages and the net gain from working. The first two roughly cancel each other. However, the correlation

change itself derives at least in part from growing inequality. Nonwage factors dominated female labor

supply decisions in the 1970's when wage inequality was relatively small. By 1990, wages had become

unequal enough that they dominated nonwage factors, so that nonworking women tended to be the ones

with less earnings potential.

Many in the literature (see Moffitt, 1999, for a survey) have concluded that selection bias may

be a relatively minor factor for understanding womens wages. Our Figure 5 shows that this may have

been the case for U.S. cross-sections sampled in the late 1970's or early 1980's (remember that the sample

made famous by Mroz, 1987, and subsequent work by Whitney et al, 1990, and others, was from 1975).

However, our Figure 5 suggests that selection bias was significant in the early 1970's and, in the other

direction, since the mid-1980's. At the very least, selection bias makes significant contributions to

measured wage growth for women and other groups with relatively weak attachments to the labor force.

Our estimates suggest that womens earnings potential has grown, if at all, far less than

previously estimated. Does this mean that female labor supply increases should not be attributed to wage

changes, but rather to social forces or technological change in the nonmarket sector (e.g., Goldin and

Gender Equality 21

Figure 8 The Prevalence of Teachers by Degree

Katz, 2002; Greenwood et al, 2001)? Answering this question is beyond the scope of this paper, but we

argue elsewhere (Mulligan and Rubinstein, 2002) that even if the average womans earnings potential

had remained constant, growing wage inequality within gender might have pulled women into the labor

force. Hence attributing female labor supply to wages versus other factors is still a topic for future

research.

VI. Appendix: The Occupational Composition of Wives with Advanced Degrees

Figure 8s solid line shows how many of the working wives with advanced degrees are teachers.

Might this bias our inference from Figure 2? On one hand, teachers have had less wage growth than

other college graduates. On the other hand, teachers earn less than other advanced degree women, and

the prevalence of teachers among working wives with advanced degrees has been declining. Also,

Figure 8s dashed line shows that the teachers are just as prevalent among working wives with college

degrees, so teachers might not bias Figure 2's advanced degree wage growth estimate relative to that for

college graduates.

Gender Equality 22

8According to the CPS, these six occupations account for about 60% of married prime-agednonteachers working full-time full-year with advanced degrees, among both women and men,throughout the period 1975-present.

Figure 9 The Advanced Degree Gender Gap by Occupation

Figure 9s solid line is the same gender gap for advanced degree women as displayed in Figure

2. The dashed line is the gender gap for nonteachers, which has essentially the same (lack of) trend.

Aside from teaching, the main occupations for advanced degree women have been (in order of

their prevalence) managers, nurses, physicians, professors, scientists, and lawyers.8 Within these six

occupations, managers gained the most advanced degree married women, but relatively few married men.

The importance and growth of the manager category by itself tends to close the advanced degree gender

gap, because managers earn somewhat more than the average advanced degree wife, and because the

gender gap among managers closed 0.08 log points. However, closure of 0.08 is much less than we see

in Figure 2 for the other education groups. Furthermore, the gender gap seems to have widened for some

occupations, such as physicians. In summary, the gender gaps by occupation have closed too little, and

Gender Equality 23

shifts of women toward high wage occupations have been too little, for the overall gender gap among

persons with advanced degrees to close significantly.

VII. References

Ahn, Hyungtaik, and James L. Powell. Semiparametric Estimation of Censored Selection Models with

a Nonparametric Selection Mechanism. Journal of Econometrics. 58(1), July 1993: 3-29.

Becker, Gary S. How the Market Acted Affirmatively for Women. Business Week. May 13, 1985.

Blau, Francine D. and Lawrence M. Kahn. Swimming Upstream: Trends in the Gender Wage

Differential in the 1980's. Journal of Labor Economics. 15(1), Part 1, January 1997: 1-42.

Chamberlain, Gary. Asymptotic Efficiency in Semiparametric Models with Censoring. Journal of

Econometrics. 32(2), July 1986: 189-218.

Das, Mitali, Whitney K. Newey, and Francis Vella. Nonparametric Estimation of Sample Selection

Models. Review of Economic Studies. 70(1), January 2003: 33-58.

Goldin, Claudia. Understanding the Gender Gap. New York: Oxford University Press, 1990.

Goldin, Claudia, and Lawrence F. Katz. The Power of the Pill: Oral Contraceptives and Womens

Career and Marriage Decisions. Journal of Political Economy. 110(4), August 2002: 730-70.

Gottschalk, Peter and Robert Moffitt. The Growth of Earnings Instability in the U.S. Labor Market.

Brookings Papers on Economic Activity. 2, 1994: 217-54.

Greenwood, Jeremy, Ananth Seshadri, and Mehmet Yorukoglu. Engines of Liberation. mimeo,

University of Rochester, 2001.

Gronau, Reuben. Wage Comparisons-A Selectivity Bias. Journal of Political Economy. 82(6),

December 1974: 1119-43.

Heckman, James J. Shadow Prices, Market Wages, and Labor Supply. Econometrica. 42(4), July

1974: 679-94.

Heckman, James J. Sample Selection Bias as a Specification Error. Econometrica. 47(1), January

1979: 153-61.

Heckman, James J. Varieties of Selection Bias. American Economic Review. 80(2), May 1990:

313-318.

Heckman, James J. and Bo E. Honore. The Empirical Content of the Roy Model. Econometrica.

58(5), September 1990: 1121-49.

Gender Equality 24

Heckman, James J. and Jose Scheinkman. The Importance of Bundling in a Gorman-Lancaster Model

of Earnings. Review of Economic Studies. 54(2), April 1987: 243-55.

Juhn, Chinhui and Dae Il Kim. The Effects of Rising Female Labor Supply on Male Wages. Journal

of Labor Economics. 17(1), January 1999: 23-48.

Juhn, Chinhui, and Kevin M. Murphy. Wage Inequality and Family Labor Supply. Journal of Labor

Economics. 15(1), Part 1, January 1997: 72-97.

Juhn, Chinhui, Kevin M. Murphy, and Brooks Pierce. Wage Inequality and the Rise in Returns to

Skill. Journal of Political Economy. 101(3), June 1993: 410-42.

Katz, Lawrence F. and David H. Autor. Changes in the Wage Structure and Earnings Inequality.

Handbook of Labor Economics. Volume 3A. New York: Elsevier Science, 1999: 1463-1555.

Katz, Lawrence F. and Kevin M. Murphy. Changes in Relative Wages, 1963-1987: Supply and

Demand Factors. Quarterly Journal of Economics. 107(1), February 1992: 35-78.

Mincer, Jacob. Labor Force Participation of Married Women: A Study of Labor Supply. in H.G.

Lewis, ed. Aspects of Labor Economics. Princeton: Princeton University Press, 1962.

Moffitt, Robert A. New Developments in Econometric Methods for Labor Market Analysis. in Orley

Ashenfelter and David Card, eds. Handbook of Labor Economics. Volume 3A. Amsterdam:

Elsevier Science, North-Holland, 1999: 1367-97.

Mroz, Thomas A. The Sensitivity of an Empirical Model of Married Women's Hours of Work to

Economic and Statistical Assumptions. Econometrica. 55(4), July 1987: 765-99.

Mulligan, Casey B. and Yona Rubinstein. Specialization, Inequality, and the Labor Market for Married

Women. manuscript, University of Chicago, July 2002.

Murphy, Kevin M. and Finis Welch. Wage Inequality in the 1990's. paper presented at the 1999

American Economic Association Meetings, January 1999.

Neal, Derek. The Measured Black-White Wage Gap among Women Is Too Small. Journal of

Political Economy. 112(1), Part 2, February 2004: S1-28.

Newey, Whitney K. Consistency of Two-Step Sample Selection Estimators Despite Misspecification

of Distribution. Economics Letters. 63(2), May 1999: 129-32.

Roy, A.D. Some Thoughts on the Distribution of Earnings. Oxford Economic Papers. 3(2), June

1951: 135-46.

Topel, Robert H. Regional Labor Markets and the Determinants of Wage Inequality. American

Economic Review. 84(2), May 1994: 17-22.